Finite Larmor radius approximation for collisional magnetic confinement. Part I: The linear Boltzmann equation
Authors:
Mihai Bostan and Céline Caldini-Queiros
Journal:
Quart. Appl. Math. 72 (2014), 323-345
MSC (2010):
Primary 35Q75, 78A35, 82D10.
DOI:
https://doi.org/10.1090/S0033-569X-2014-01356-1
Published electronically:
March 28, 2014
MathSciNet review:
3186240
Full-text PDF Free Access
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Additional Information
Abstract: The subject matter of this paper concerns the derivation of the finite Larmor radius approximation, when collisions are taken into account. Several studies are performed, corresponding to different collision kernels : the relaxation and the Fokker-Planck operators. Gyroaveraging the relaxation operator leads to a position-velocity integral operator, whereas gyroaveraging the linear Fokker-Planck operator leads to diffusion in velocity but also with respect to the perpendicular position coordinates.
References
- Mihai Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal. 61 (2009), no. 2, 91–123. MR 2499194
- Mihai Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations 249 (2010), no. 7, 1620–1663. MR 2677810, DOI https://doi.org/10.1016/j.jde.2010.07.010
- Mihai Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, Multiscale Model. Simul. 8 (2010), no. 5, 1923–1957. MR 2769087, DOI https://doi.org/10.1137/090777621
- M. Bostan, Transport of charged particles under fast oscillating magnetic fields, SIAM J. Math. Anal. 44(2012) 1415-1447.
- Mihaï Bostan and Céline Caldini-Queiros, Approximation de rayon de Larmor fini pour les plasmas magnétisés collisionnels, C. R. Math. Acad. Sci. Paris 350 (2012), no. 19-20, 879–884 (French, with English and French summaries). MR 2990896, DOI https://doi.org/10.1016/j.crma.2012.09.019
- M. Bostan, C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part II : The Fokker-Planck-Landau equation, to appear in Quart. Appl. Math.
- Alain J. Brizard, Variational principle for nonlinear gyrokinetic Vlasov-Maxwell equations, Phys. Plasmas 7 (2000), no. 12, 4816–4822. MR 1800518, DOI https://doi.org/10.1063/1.1322063
- Alain J. Brizard, A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields, Phys. Plasmas 11 (2004), no. 9, 4429–4438. MR 2095562, DOI https://doi.org/10.1063/1.1780532
- A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys. 79 (2007), no. 2, 421–468. MR 2336960, DOI https://doi.org/10.1103/RevModPhys.79.421
- E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal. 46 (2006), no. 1, 1–28. MR 2196630
- E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math. Adv. Appl. 4 (2010), no. 2, 135–169. MR 2816864
- Emmanuel Frénod and Eric Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal. 18 (1998), no. 3-4, 193–213. MR 1668938
- Emmanuel Frénod and Eric Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal. 32 (2001), no. 6, 1227–1247. MR 1856246, DOI https://doi.org/10.1137/S0036141099364243
- X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard, Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations, Phys. Plasmas 16(2009).
- François Golse and Laure Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9) 78 (1999), no. 8, 791–817. MR 1715342, DOI https://doi.org/10.1016/S0021-7824%2899%2900021-5
- P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 1990. MR 1063852
- Frédéric Poupaud and Christian Schmeiser, Charge transport in semiconductors with degeneracy effects, Math. Methods Appl. Sci. 14 (1991), no. 5, 301–318. MR 1113606, DOI https://doi.org/10.1002/mma.1670140503
- F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech. 72 (1992), no. 8, 359–372 (English, with English, German and Russian summaries). MR 1178932, DOI https://doi.org/10.1002/zamm.19920720813
- Laure Saint-Raymond, Control of large velocities in the two-dimensional gyrokinetic approximation, J. Math. Pures Appl. (9) 81 (2002), no. 4, 379–399. MR 1967354, DOI https://doi.org/10.1016/S0021-7824%2801%2901245-4
- X. Q. Xu, M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids, B 3(1991) 627-643.
References
- Mihai Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal. 61 (2009), no. 2, 91–123. MR 2499194 (2009k:82122)
- Mihai Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations 249 (2010), no. 7, 1620–1663. MR 2677810 (2011i:82074), DOI https://doi.org/10.1016/j.jde.2010.07.010
- Mihai Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, Multiscale Model. Simul. 8 (2010), no. 5, 1923–1957. MR 2769087 (2012b:35342), DOI https://doi.org/10.1137/090777621
- M. Bostan, Transport of charged particles under fast oscillating magnetic fields, SIAM J. Math. Anal. 44(2012) 1415-1447.
- Mihaï Bostan and Céline Caldini-Queiros, Approximation de rayon de Larmor fini pour les plasmas magnétisés collisionnels, C. R. Math. Acad. Sci. Paris 350 (2012), no. 19-20, 879–884. MR 2990896, DOI https://doi.org/10.1016/j.crma.2012.09.019
- M. Bostan, C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part II : The Fokker-Planck-Landau equation, to appear in Quart. Appl. Math.
- Alain J. Brizard, Variational principle for nonlinear gyrokinetic Vlasov-Maxwell equations, Phys. Plasmas 7 (2000), no. 12, 4816–4822. MR 1800518 (2001g:76065), DOI https://doi.org/10.1063/1.1322063
- Alain J. Brizard, A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields, Phys. Plasmas 11 (2004), no. 9, 4429–4438. MR 2095562 (2005e:82094), DOI https://doi.org/10.1063/1.1780532
- A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys. 79 (2007), no. 2, 421–468. MR 2336960 (2008e:76188), DOI https://doi.org/10.1103/RevModPhys.79.421
- E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal. 46 (2006), no. 1, 1–28. MR 2196630 (2007a:34090)
- E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math. Adv. Appl. 4 (2010), no. 2, 135–169. MR 2816864 (2012d:76146)
- Emmanuel Frénod and Eric Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal. 18 (1998), no. 3-4, 193–213. MR 1668938 (99m:82046)
- Emmanuel Frénod and Eric Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal. 32 (2001), no. 6, 1227–1247 (electronic). MR 1856246 (2002g:82049), DOI https://doi.org/10.1137/S0036141099364243
- X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard, Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations, Phys. Plasmas 16(2009).
- François Golse and Laure Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9) 78 (1999), no. 8, 791–817. MR 1715342 (2000g:35209), DOI https://doi.org/10.1016/S0021-7824%2899%2900021-5
- P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 1990. MR 1063852 (91j:78011)
- Frédéric Poupaud and Christian Schmeiser, Charge transport in semiconductors with degeneracy effects, Math. Methods Appl. Sci. 14 (1991), no. 5, 301–318. MR 1113606 (92c:82102), DOI https://doi.org/10.1002/mma.1670140503
- F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech. 72 (1992), no. 8, 359–372 (English, with English, German and Russian summaries). MR 1178932 (93h:82074), DOI https://doi.org/10.1002/zamm.19920720813
- Laure Saint-Raymond, Control of large velocities in the two-dimensional gyrokinetic approximation, J. Math. Pures Appl. (9) 81 (2002), no. 4, 379–399. MR 1967354 (2004b:76166), DOI https://doi.org/10.1016/S0021-7824%2801%2901245-4
- X. Q. Xu, M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids, B 3(1991) 627-643.
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Additional Information
Mihai Bostan
Affiliation:
Laboratoire d’Analyse, Topologie, Probabilités LATP, Centre de Mathématiques et Informatique CMI, UMR CNRS 7353, 39 rue Frédéric Joliot Curie, 13453 Marseille Cedex 13 France
Email:
bostan@cmi.univ-mrs.fr
Céline Caldini-Queiros
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex France
Email:
celine.caldini-queiros@univ-fcomte.fr
Received by editor(s):
July 4, 2012
Received by editor(s) in revised form:
December 15, 2012
Published electronically:
March 28, 2014
Article copyright:
© Copyright 2014
Brown University